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Theorem iooval2 13331

Description: Value of the open interval function. (Contributed by NM, 6-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)

**Assertion**
Ref	Expression
iooval2	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴(,)𝐵) = {𝑥 ∈ ℝ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)})

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵

**Proof of Theorem iooval2**
Step	Hyp	Ref	Expression
1		iooval 13322	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴(,)𝐵) = {𝑥 ∈ ℝ^ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)})
2		elioore 13328	. . . . . 6 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ∈ ℝ)
3	2	ssriv 3925	. . . . 5 ⊢ (𝐴(,)𝐵) ⊆ ℝ
4	1, 3	eqsstrrdi 3967	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → {𝑥 ∈ ℝ^ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)} ⊆ ℝ)
5		dfss2 3907	. . . 4 ⊢ ({𝑥 ∈ ℝ^* ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)} ⊆ ℝ ↔ ({𝑥 ∈ ℝ^* ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)} ∩ ℝ) = {𝑥 ∈ ℝ^* ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)})
6	4, 5	sylib 218	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ({𝑥 ∈ ℝ^ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)} ∩ ℝ) = {𝑥 ∈ ℝ^* ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)})
7		inrab2 4257	. . . 4 ⊢ ({𝑥 ∈ ℝ^* ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)} ∩ ℝ) = {𝑥 ∈ (ℝ^* ∩ ℝ) ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)}
8		ressxr 11189	. . . . . 6 ⊢ ℝ ⊆ ℝ^*
9		sseqin2 4163	. . . . . 6 ⊢ (ℝ ⊆ ℝ^* ↔ (ℝ^* ∩ ℝ) = ℝ)
10	8, 9	mpbi 230	. . . . 5 ⊢ (ℝ^* ∩ ℝ) = ℝ
11	10	rabeqi 3402	. . . 4 ⊢ {𝑥 ∈ (ℝ^* ∩ ℝ) ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)} = {𝑥 ∈ ℝ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)}
12	7, 11	eqtri 2759	. . 3 ⊢ ({𝑥 ∈ ℝ^* ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)} ∩ ℝ) = {𝑥 ∈ ℝ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)}
13	6, 12	eqtr3di 2786	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → {𝑥 ∈ ℝ^ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)} = {𝑥 ∈ ℝ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)})
14	1, 13	eqtrd 2771	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴(,)𝐵) = {𝑥 ∈ ℝ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 {crab 3389 ∩ cin 3888 ⊆ wss 3889 class class class wbr 5085 (class class class)co 7367 ℝcr 11037 ℝ^*cxr 11178 < clt 11179 (,)cioo 13298

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-pre-lttri 11112 ax-pre-lttrn 11113

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-id 5526 df-po 5539 df-so 5540 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-ov 7370 df-oprab 7371 df-mpo 7372 df-1st 7942 df-2nd 7943 df-er 8643 df-en 8894 df-dom 8895 df-sdom 8896 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-ioo 13302

This theorem is referenced by: elioo2 13339 ioomax 13375 ioopos 13377 dfioo2 13403

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